QM-AM-GM-HM Inequality

Let \(a_1, ..., a_N\) be a set of \(N\) positive numbers.

Define the four following means:

1. Arithmetic:

\[\mu_{AM} = \frac{1}{N} \sum_n a_n\]

1. Quadratic:

\[\mu_{QM} = \Big( \frac{1}{N} \sum_n a_n^2 \Big)^2\]

1. Geometric:

\[\mu_{GM} = \Big( \prod_n a_n \Big)^N\]

1. Harmonic Mean:

\[\mu_{HM} = \frac{N}{\sum_n \frac{1}{a_n}}\]

The QM-AM-GM-HM Inequality states that

\[\mu_{QM} \geq \mu_{AM} \geq \mu_{GM} \geq \mu_{HM}\]

Inclusion of Power Mean

One can generalize the arithmetic and quadratic means to the power mean, which is defined as:

\[\mu_{PM}^{(k)} = \Big( \frac{1}{N} \sum a_n^k \Big)^{1/N}\]

For two powers \(k_1> k_2 > 0\), the power mean inequality states that

\[\mu_{PM}^{(k_1)} \geq \mu_{PM}^{(k_2)}\]

This includes the above QM-AM relationship, since \(\mu_{QM} = \mu_{PM}^{(2)}\) and \(\mu_{AM} = \mu_{PM}^{(1)}\).